Eigenstate correlations around many-body localization transition

TL;DR

This paper investigates eigenstate correlations near the many-body localization transition in spin chains and random graphs, revealing power-law behaviors and a maximum correlation at the transition point, with implications for quantum simulations.

Contribution

It compares eigenstate correlations in the MBL transition for spin chains and random regular graphs, highlighting universal behaviors and potential measurement methods on quantum devices.

Findings

Eigenstate correlation function shows power-law enhancement on the localized side.

Maximum correlation occurs at the transition point Wc.

Correlation functions relate to Hilbert-space return probability.

Abstract

We explore correlations of eigenstates around the many-body localization (MBL) transition in their dependence on the energy difference (frequency) $\omega$ and disorder $W$. In addition to the genuine many-body problem, XXZ spin chain in random field, we consider localization on random regular graphs (RRG) that serves as a toy model of the MBL transition. Both models show a very similar behavior. On the localized side of the transition, the eigenstate correlation function $\beta(\omega)$ shows a power-law enhancement of correlations with lowering $\omega$; the corresponding exponent depends on $W$. The correlation between adjacent-in-energy eigenstates exhibits a maximum at the transition point $W_c$, visualizing the drift of $W_c$ with increasing system size towards its thermodynamic-limit value. The correlation function $\beta(\omega)$ is related, via Fourier transformation, to the Hilbert-space return probability. We discuss measurement of such (and related) eigenstate correlation functions on state-of-the-art quantum computers and simulators.